Rates of Convergence of Newton's Method.

Given an operator P in a Banach space X with Lipschitz continuous derivative P primed, it is shown that the existence of 1P primed x 1 is necessary and sufficient to predict on the basis of the theorem of L. V. Kantorovic that the Newton sequence x sub n 1 x sub n - Px sub nP primeds sub n will converge to a solution x of the equation Px o quadratically. Some examples are given of convergent Newton sequences for which convergence and the rate of convergence cannot be predicted by the Kantorovic theorem. Author